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Modeling latent spatio-temporal disease incidence using penalized composite link models
Get PDFEpidemiological data are frequently recorded at coarse spatio-temporal resolutions to protect confidential information or to summarize it in a compact manner. However, the detailed patterns followed by the source data, which may be of interest to researchers and public health officials, are overlooked. We propose to use the penalized composite link model (Eilers PCH (2007)), combined with spatio-temporal P-splines methodology (Lee D.-J., Durban M (2011)) to estimate the underlying trend within data that have been aggregated not only in space, but also in time. Model estimation is carried out within a generalized linear mixed model framework, and sophisticated algorithms are used to speed up computations that otherwise would be unfeasible. The model is then used to analyze data obtained during the largest outbreak of Q-fever in the Netherlands.Grant No. MTM2014-52184-P awarded to MD, and DA, and by Agencia Estatal de Investigació
Entire vortex solutions of negative degree for the anisotropic Ginzburg-Landau system
Get PDFThe anisotropic Ginzburg-Landau system , for and , models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that and has a prescribed topological degree as , for small values of the anisotropy parameter . Unlike the isotropic case , this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a finite symmetry subgroup.FONDECYT 1210405, Chilean research grant,
France-Chile ECOS-Sud C18E06 and ANID projects ACE210010 and FB210005,
ANR project ANR-18-CE40-0023 and COOPINTER project IEA-297303.
ANID projects ACE210010 and FB210005.
National Science Centre, Poland (Grant No. 2017/26/E/ST1/00817).
BERC 2018-2021 program.
BCAM Severo Ochoa excellence accreditation SEV-2017-0718.
Project PID2020-114189RB-I00 (PID2020-114189RB-I00 / AEI / 10.13039/501100011033)
Spectral analysis of Dirac operators on bounded domains
Get PDFThis thesis is devoted to the spectral study of two types of perturbation of the
Dirac operator, which are singular from the point of view of scaling.
In the first part of this thesis, we consider the coupling of the Dirac operator with a combi-
nation of delta-shell interactions of electrostatic, Lorentz scalar, and magnetic type supported
either on regular compact surfaces or locally deformed hyperplanes. We develop an approach
based on regularization techniques that will allow us to describe the self-adjoint realization
of the perturbed Dirac operator for any combination of the coupling constants. We then in-
vestigate the qualitative spectral properties of the various models using a Birman-Schwinger
principle and a Krein-type formula relating the resolvent of the perturbed operator to that
of the free Dirac operator, and we pay special attention to the case of critical combinations
of coupling constants and those that give rise to the phenomenon of confinement.
In the second part, we study the coupling of the Dirac operator with non-critical combi-
nations of delta interactions supported on non-regular compact surfaces. We first generalize
the results obtained in the context of regular surfaces to the case of surfaces locally coincident
with the graph of a Lipschitz function whose gradient is bounded and has vanishing mean
oscillations. For this we use some techniques from harmonic analysis, potential theory and
Fredholm’s theory. Moreover, in the case of Hölder surfaces, we show how the smoothness
of the surface supporting the delta interactions affects the Sobolev regularity of the domain
of the operator under consideration. In a second step, we investigate delta-interactions sup-
ported on surfaces satisfying certain weak topological conditions. We first study the Dirac
operator coupled with the electrostatic and Lorentz scalar delta-shell interactions supported
on uniformly rectifiable surfaces. Under certain conditions on the coupling constants, we
prove the self-adjointness fo the perturbed operator and we establish several spectral proper-
ties in the Lipschitz case. In particular, we determine the essential spectrum of the perturbed
operator and we show that at most a finite number of eigenvalues can appear in the gap.
Moreover, we fit these results to other delta-shell interactions and derive several models of
Dirac operators that give rise to the confinement phenomenon.
In the third part of this thesis, we are concerned the study of the pseudodifferential
properties of Poincaré-Steklov (PS) operators associated with the Dirac operator with the
MIT bag boundary condition. First, we show that the PS operators fit well into the framework
of classical pseudodifferential operators. Then, we study the PS operators from the point
of view of semiclassical pseudodifferential operators, where the semiclassical parameter is
given by the inverse of the mass. In particular, using some regularity properties of the MIT
bag operator, we show that the PS operators are zero-order semiclassical pseudodifferential
operators, and we determine their semiclassical principal symbols. In a second step, we study
the Dirac operator coupled with a potential depending on an additional mass and supported
outside a regular domain. When the additional mass is large enough, using the symbolic
calculus and the properties of the PS operators, we establish a Krein-type formula relating
the resolvent of the perturbed operator to that of the MIT bag operator. With its help, we
show that the perturbed operator converges in the norm resolvent sense towards the MIT
bag operator and give a sharp estimate of the convergence rate
Extended State Space for Describing Renormalized Fock Spaces in QFT
Get PDFIn quantum field theory (QFT) models, it often seems natural to use, instead of wave functions from Fock space, wave functions that are not square-integrable and have prefactors involving divergent integrals (known as infinite wave function renormalizations). Here, we rigorously construct vector spaces containing divergent integrals, as well as two extended state vector spaces, which contain a dense subspace of Fock space, but also incorporate non-square-integrable wave functions with infinite wave function renormalizations.
As a demonstration, we apply this construction to a non-perturbative renormalization of a class of simple non-relativistic QFT models, which are polaron models with resting fermions. The Hamiltonian without cutoffs, an infinite self-energy and a dressing transformation are defined as linear operators on certain subspaces of the two Fock space extensions. This way, we can obtain a renormalized Hamiltonian which can be realized as a densely defined self-adjoint operator on Fock space.Wilhelm Schuler-Stiftung Tübingen,
DAAD (Deutscher Akademischer Austauschdienst
Self-diffusion of spherocylindrical particles flowing under non-uniform shear rate
Get PDFThis work is devoted to study numerically the self-diffusion of spherocylindrical particles flowing down an inclined plane, using the discrete element method (DEM). This system is challenging due to particles being non-spherical and because they are subjected to a non-uniform shear rate. We performed simulations for several aspect ratios and inclination angles, tracking individual particle trajectories. Using the simulation data, we computed the diffusion coefficients D, and a coarse-graining methodology allowed accessing the shear rate spatial profiles(z). This data enabled us to identify the spatial regions where the diffusivity strongly correlates with the local shear rate. Introducing an effective particle size d⊥, we proposed a well-rationalized scaling law between D and . Our findings also identified specific locations where the diffusivity does not correlate with the shear rate. This observation corresponds to zones where has non-linear spatial variation, and the velocity probability density distributions exhibit asymmetric shapes
Fock-space approach to stochastic susceptible-infected-recovered models
No full textWe investigate the stochastic susceptible-infected-recovered (SIR) model of infectious disease dynamics in the Fock-space approach. In contrast to conventional SIR models based on ordinary differential equations for the subpopulation sizes of S, I, and R individuals, the stochastic SIR model is driven by a master equation governing the transition probabilities among the system’s states defined by SIR occupation numbers. In the Fock-space approach the master equation is recast in the form of a real-valued Schrödinger-type equation with a second quantization Hamiltonian-like operator describing the infection and recovery processes. We find exact analytic expressions for the Hamiltonian eigenvalues for any population size N. We present small- and large-N results for the average numbers of SIR individuals and basic reproduction number. For small N we also obtain the probability distributions of SIR states, epidemic sizes and durations, which cannot be found from deterministic SIR models. Our Fock-space approach to stochastic SIR models introduces a powerful set of tools to calculate central quantities of epidemic processes, especially for relatively small populations where statistical fluctuations not captured by conventional deterministic SIR models play a crucial role.We investigate the stochastic susceptible-infected-recovered (SIR) model of infectious disease dynamics in the Fock-space approach. In contrast to conventional SIR models based on ordinary differential equations for the subpopulation sizes of S, I, and R individuals, the stochastic SIR model is driven by a master equation governing the transition probabilities among the system’s states defined by SIR occupation numbers. In the Fock-space approach the master equation is recast in the form of a real-valued Schrödinger-type equation with a second quantization Hamiltonian-like operator describing the infection and recovery processes. We find exact analytic expressions for the Hamiltonian eigenvalues for any population size N. We present small- and large-N results for the average numbers of SIR individuals and basic reproduction number. For small N we also obtain the probability distributions of SIR states, epidemic sizes and durations, which cannot be found from deterministic SIR models. Our Fock-space approach to stochastic SIR models introduces a powerful set of tools to calculate central quantities of epidemic processes, especially for relatively small populations where statistical fluctuations not captured by conventional deterministic SIR models play a crucial role
Magnetic Reconnection in Magnetohydrodynamics
No full textWe provide examples of periodic solutions (in both 2 and 3 dimension) of the Magnetohydrodynamics equations such that the topology of the magnetic lines changes during the evolution. This phenomenon, known as magnetic reconnection, is relevant for physicists, in particular in the study of highly conducting plasmas. Although numerical and experimental evidences exist, analytical examples of magnetic reconnection were not known
Damage identification in bridges combining deep learning and computational mechanic
No full textCivil infrastructures, such as bridges, are critical assets for society and the economy. Many of them have already reached their expected life and withstand loadings that exceed the design specifications. Besides, bridges suffer from various degradation mechanisms, including aging, corrosion, earthquakes, and, nowadays, the undeniable effect of climate change. This context has motivated an increasing interest in early detecting damage to prevent costly actions and dangerous failures. Structural Health Monitoring (SHM) consists of implementing effective strategies to continuously assess the health condition of structures using monitoring data collected by sensors. This dissertation focuses on the SHM problem of damage detection and identification. It is an ill-posed inverse problem that aims at inferring the health state of a structure from measurements of its response. The measurements include large amounts of noisy data affected by environmental and operational conditions, acquired with sensors of different nature. Solving such a multidisciplinary problem encompasses the use of applied mathematics, computational mechanics, and data science. In this dissertation, we exploit the potential of Deep Neural Networks in approximating complex inverse problems and employ computational parametrizations and the Finite Element Method to enrich the training phase by including damage scenarios. We explore two different approaches to the problem. In the first approach, we develop an outlier detection strategy to detect departures from the baseline condition. We only employ long-term monitoring data measured at the bridge during normal (healthy) operation. Starting from Principal Component Analysis (PCA) as a statistical data reconstruction technique, we design a specific Deep Autoencoder network that enhances PCA by adding residual connections to include nonlinear transformations. This architecture gains partial explainability by evaluating the contribution of nonlinearties over affine transformations in the reconstruction process. We also investigate the method performance when using local or global variables and evaluate the potential of combining both data sources in the damage detection task. In the second approach, we reach a higher level of damage identification by estimating its severity and location. The goal is to provide a suitable methodology for real full-scale applications that requires reasonable computational resources. We employ a calibrated computational parametrization to solve multiple Finite Element simulations under different damage scenarios. These synthetic scenarios enrich the training dataset of a Deep Neural Network that maps the response of the bridge with its health condition in terms of damage location and severity. Finally, we incorporate the effect of environmental and operational variability in the parametrization by applying a clustering algorithm to find representative samples among the entire dataset. We assume these samples cover most of the variability present in the data and consider them as starting points to generate synthetic training data. We apply the proposed methods to three main case study bridges with available monitoring data: the Beltran bridge in Mexico, and the Infante Dom Henrique bridge in Porto, and the Z24 bridge in Switzerland. Both structures resulted critical to validate and test the ability of the proposed methods and to demonstrate their applicability in the full-scale.This disseration has been possible thanks to the support received from: the European Union’s Horizon 2020 research and innovation program under the grant agreement No 769373 (FORESEE project) and the Marie Sklodowska-Curie grant agreement No 777778 (MATHROCKS); the Base Funding - UIDB/04708/2020 of the CONSTRUCT - Instituto de I&D em Estruturas e Constru¸c˜oes - funded by national funds through the FCT/MCTES (PIDDAC); the European Regional Development Fund (ERDF) through the Interreg V-A Spain-France-Andorra program POCTEFA 2014-2020 Project PIXIL (EFA362/19); the Spanish Ministry of Science and Innovation with references PID2019-108111RB-I00 (FEDER/AEI) and the “BCAM Severo Ochoa” accreditation of excellence (SEV-2017-0718); and the Basque Government through the BERC 2018-2021 program, the four Elkartek projects 3KIA (KK-2020/00049), EXPERTIA (KK-2021/00048), MATHEO (KK-2019-00085), and SIGZE (KK-2021/00095); the grant “Artificial Intelligence in BCAM number EXP. 2019/00432”, and the Consolidated Research Group MATHMODE (IT1294-19) given by the Department of Education
Motion of a rigid body in a compressible fluid with Navier-slip boundary condition
Get PDFIn this work, we study the motion of a rigid body in a bounded domain which is filled with a compressible isentropic fluid. We consider the Navier-slip boundary condition at the interface as well as at the boundary of the domain. This is the first mathematical analysis of a compressible fluid-rigid body system where Navier-slip boundary conditions are considered. We prove existence of a weak solution of the fluid-structure system up to collision
The Fokker–Planck equation of the superstatistical fractional Brownian motion with application to passive tracers inside cytoplasm
Get PDFBy collecting from literature data experimental evidence of anomalous diffusion of passive tracers inside cytoplasm, and in particular of subdiffusion of mRNA molecules inside live Escherichia coli cells, we obtain the probability density function of molecules’ displacement and we derive the corresponding Fokker–Planck equation. Molecules’ distribution emerges to be related to the Krätzel function and its Fokker–Planck equation to be a fractional diffusion equation in the Erdélyi–Kober sense. The irreducibility of the derived Fokker–Planck equation to those of other literature models is also discussed.BERC 2018–2021
BERC 2022–202